Optimal Renormalization Group Transformation from Information Theory

TL;DR

This paper explores an information-theoretic approach to real-space renormalization group (RG) transformations, demonstrating their optimality in controlling interaction range and correlations, with empirical validation on Ising chains.

Contribution

It proves that optimal RSMI-based RG does not increase interaction range and suppresses correlation growth, linking RG properties to physical Hamiltonians and disorder distributions.

Findings

Perfect RSMI coarse-graining limits interaction range.

Optimal RG suppresses correlation growth in disordered systems.

Empirical validation on Ising chains confirms theoretical predictions.

Abstract

Recently a novel real-space RG algorithm was introduced, identifying the relevant degrees of freedom of a system by maximizing an information-theoretic quantity, the real-space mutual information (RSMI), with machine learning methods. Motivated by this, we investigate the information theoretic properties of coarse-graining procedures, for both translationally invariant and disordered systems. We prove that a perfect RSMI coarse-graining does not increase the range of interactions in the renormalized Hamiltonian, and, for disordered systems, suppresses generation of correlations in the renormalized disorder distribution, being in this sense optimal. We empirically verify decay of those measures of complexity, as a function of information retained by the RG, on the examples of arbitrary coarse-grainings of the clean and random Ising chain. The results establish a direct and quantifiable connection between properties of RG viewed as a compression scheme, and those of physical objects i.e. Hamiltonians and disorder distributions. We also study the effect of constraints on the number and type of coarse-grained degrees of freedom on a generic RG procedure.